A215733 -id:A215733 - cp's OEIS frontend

A215732 a(n) is the first digit to appear n times in succession in a power of 2.

1, 5, 7, 5, 6, 8, 7, 1, 9, 9, 6, 3, 2, 9, 1, 4, 1

Offset: 1

V. Raman, Aug 22 2012

Assumes that there are no 2 digits that appear n times in succession for the first time in the same power of 2. - Chai Wah Wu, Apr 26 2019

			2^16 = 65536 is the first power of 2 with a repeated digit (cf. A045875), with 5 repeated, so a(2) = 5. - _N. J. A. Sloane_, Aug 23 2012

Cf. A045875, A215733, A215734, A215735.

n = 1; x = 1; lst = {};
For[i = 1, i <= 10000, i++,
z = Split[IntegerDigits[x]]; a = Length /@ z; b = Max[a];
For[j = n, j <= b, j++,
  AppendTo[lst, First[First[Part[z, First[Position[a, b]]]]]]; n++
]; x = 2 x ]; lst (* Robert Price, Mar 16 2019 *)

def A215732(n):
    l, x = [str(d)*n for d in range(10)], 1
    for m in range(10**9):
        s = str(x)
        for k in range(10):
            if l[k] in s:
                return k
        x *= 2
    return 'search limit reached'
# Chai Wah Wu, Dec 17 2014

a(11)-a(13) added by T. D. Noe, Sep 04 2012
a(14) added by T. D. Noe, Sep 06 2012
a(15) from Bert Dobbelaere, Feb 25 2019
a(16) from Paul Geneau de Lamarlière, Jun 26 2024
a(17) from Paul Geneau de Lamarlière, Sep 24 2024

A215727 a(n) is the smallest m for which 3^m contains n consecutive identical digits.

Original entry on oeis.org

0, 11, 32, 33, 274, 538, 2124, 7720, 22791, 107187, 107187, 639226, 5756979, 8885853, 68353787, 78927180, 78927180

Offset: 1

V. Raman, Aug 22 2012

3^(a(16)+1) contains exactly 16 consecutive 3's. - Bert Dobbelaere, Mar 20 2019

			3^11 = 177147, which has two digits in a row.

Cf. A215733, A045875.

A215727[n_] := Module[{m = 0 , t}, t = Table[i, {i, 0, 9}, {n}];
While[True, If[ContainsAny[Subsequences[IntegerDigits[3^m], {n}], t], Return[m], m++]]; m]; Table[A215727[n], {n, 1, 14}] (* Robert Price, Oct 16 2018 *)

def A215727(n):
    l, x = [str(d)*n for d in range(10)], 1
    for m in range(10**9):
        s = str(x)
        for k in l:
            if k in s:
                return m
        x *= 3
    return 'search limit reached'
# Chai Wah Wu, Dec 17 2014

a(12) from Chai Wah Wu, Dec 17 2014
a(13)-a(14) from Giovanni Resta, Apr 20 2016
a(15) from Bert Dobbelaere, Mar 04 2019
a(16)-a(17) from Bert Dobbelaere, Mar 20 2019

A217171 a(n) is the first digit (from the left) to appear six times in succession in the decimal representation n^A217161(n).

Original entry on oeis.org

8, 8, 7, 2, 1, 7, 7, 8, 0, 0, 0, 2, 4, 4, 7, 4, 6, 1, 0, 2, 9, 4, 4, 5, 9, 0, 7, 4, 0, 2, 3, 9, 4, 6, 8, 0, 3, 0, 0, 0, 6, 1, 1, 5, 3, 1, 6, 0, 0, 9, 0, 2, 7, 8, 4, 7, 6, 9, 0, 2, 5, 7, 7, 5, 9, 6, 4, 7, 0, 3, 4, 0, 4, 7, 3, 1, 2, 3, 0, 0, 0, 5, 1, 2, 3, 2, 1

Offset: 2

V. Raman, Sep 27 2012

V. Raman, Table of n, a(n) for n = 2..10000

Cf. A217161.

Cf. A215732, A215733, A215734, A215735, A215736, A215737.

Table[k = 1; While[d = IntegerDigits[n^k]; df = Partition[Differences[d], 5, 1]; ! MemberQ[df, {0, 0, 0, 0, 0}], k++]; d[[Position[df, {0, 0, 0, 0, 0}][[1, 1]]]], {n, 2, 100}] (* T. D. Noe, Oct 02 2012 *)

A217167 a(n) is the first digit (from the left) to appear two times in succession in the decimal representation of n^A217157(n).

Original entry on oeis.org

5, 7, 5, 8, 7, 1, 4, 4, 0, 1, 4, 4, 4, 2, 5, 3, 8, 9, 0, 4, 2, 8, 3, 4, 1, 4, 7, 1, 0, 8, 3, 3, 1, 2, 6, 7, 4, 4, 0, 1, 8, 8, 4, 1, 1, 2, 1, 1, 0, 7, 1, 8, 1, 5, 4, 5, 3, 1, 0, 2, 4, 0, 4, 2, 6, 4, 4, 2, 0, 1, 0, 3, 2, 7, 7, 7, 5, 0, 0, 4, 5, 8, 1, 2, 0, 3, 8

Offset: 2

V. Raman, Sep 27 2012

Note that 109^2 = 11881. So by looking at the digits from the left, we have a(109) = 1.

V. Raman, Table of n, a(n) for n = 2..10000

Cf. A217157.
Cf. A215732, A215236.
Cf. A215733, A215734, A215735, A215736, A215737.

Table[k = 1; While[d = IntegerDigits[n^k]; ! MemberQ[df = Differences[d], 0], k++]; d[[Position[df, 0][[1, 1]]]], {n, 2, 100}] (* T. D. Noe, Oct 02 2012 *)

A217168 a(n) is the first digit (from the left) to appear three times in succession in the decimal representation of n^A217158(n).

Original entry on oeis.org

7, 8, 7, 8, 7, 7, 7, 8, 0, 8, 1, 6, 1, 2, 7, 2, 2, 7, 0, 2, 8, 6, 7, 8, 4, 5, 6, 1, 0, 1, 7, 4, 7, 5, 6, 3, 4, 5, 0, 5, 1, 1, 7, 4, 5, 6, 5, 1, 0, 8, 5, 1, 1, 3, 0, 1, 0, 1, 0, 3, 8, 4, 7, 2, 9, 6, 8, 7, 0, 2, 2, 8, 2, 7, 3, 4, 5, 5, 0, 8, 7, 0, 7, 5, 2, 7, 2

Offset: 2

V. Raman, Sep 27 2012

V. Raman, Table of n, a(n) for n = 2..10000

Cf. A217158.

Cf. A215732, A215733, A215734, A215735, A215736, A215737.

Table[k = 1; While[d = IntegerDigits[n^k]; ! MemberQ[df = Partition[Differences[d], 2, 1], {0, 0}], k++]; d[[Position[df, {0, 0}][[1, 1]]]], {n, 2, 100}] (* T. D. Noe, Oct 02 2012 *)

A217169 a(n) is the first digit (from the left) to appear four times in succession in the decimal representation of n^A217159(n).

Original entry on oeis.org

5, 5, 8, 7, 0, 9, 8, 5, 0, 7, 3, 6, 1, 2, 8, 9, 5, 1, 0, 4, 4, 7, 7, 8, 2, 5, 1, 0, 0, 9, 8, 6, 5, 5, 2, 6, 5, 6, 0, 9, 6, 7, 6, 5, 1, 6, 2, 2, 0, 6, 6, 3, 3, 1, 0, 3, 1, 1, 0, 7, 6, 2, 8, 3, 5, 9, 6, 8, 0, 1, 4, 8, 6, 3, 8, 1, 7, 9, 0, 5, 6, 9, 1, 2, 8, 1, 8

Offset: 2

V. Raman, Sep 27 2012

V. Raman, Table of n, a(n) for n = 2..10000

Cf. A217159.

Cf. A215732, A215733, A215734, A215735, A215736, A215737.

Table[k = 1; While[d = IntegerDigits[n^k]; df = Partition[Differences[d], 3, 1]; ! MemberQ[df, {0, 0, 0}], k++]; d[[Position[df, {0, 0, 0}][[1, 1]]]], {n, 2, 100}] (* T. D. Noe, Oct 02 2012 *)

A217170 a(n) is the first digit (from the left) to appear five times in succession in the decimal representation of n^A217160(n).

Original entry on oeis.org

6, 5, 6, 4, 2, 5, 4, 5, 0, 6, 5, 4, 4, 1, 6, 5, 5, 9, 0, 2, 8, 6, 2, 5, 9, 2, 1, 8, 0, 2, 6, 8, 2, 6, 2, 6, 3, 8, 0, 8, 7, 0, 1, 7, 6, 3, 6, 5, 0, 6, 9, 6, 6, 9, 2, 2, 4, 4, 0, 4, 9, 4, 2, 3, 4, 4, 8, 5, 0, 2, 9, 9, 0, 9, 9, 0, 9, 6, 0, 0, 4, 9, 1, 0, 6, 1, 2

Offset: 2

V. Raman, Sep 27 2012

V. Raman, Table of n, a(n) for n = 2..10000

Cf. A217160.

Cf. A215732, A215733, A215734, A215735, A215736, A215737.

Table[k = 1; While[d = IntegerDigits[n^k]; df = Partition[Differences[d], 4, 1]; ! MemberQ[df, {0, 0, 0, 0}], k++]; d[[Position[df, {0, 0, 0, 0}][[1, 1]]]], {n, 2, 100}] (* T. D. Noe, Oct 02 2012 *)

A217172 a(n) is the first digit (from the left) to appear seven times in succession in the decimal representation of n^A217162(n).

Original entry on oeis.org

7, 2, 7, 8, 2, 3, 7, 2, 0, 6, 4, 6, 3, 2, 7, 9, 4, 9, 0, 4, 2, 4, 5, 8, 4, 2, 5, 6, 0, 2, 2, 3, 8, 7, 2, 5, 3, 0, 0, 4, 1, 9, 2, 7, 0, 8, 1, 6, 0, 4, 4, 2, 1, 2, 3, 1, 3, 4, 0, 2, 8, 8, 7, 5, 1, 7, 6, 9, 0, 4, 8, 1, 7, 5, 9, 6, 4, 3, 0, 2, 1, 1, 7, 8, 7, 8, 8

Offset: 2

V. Raman, Sep 27 2012

V. Raman, Table of n, a(n) for n = 2..1000

Cf. A217162.

Cf. A215732, A215733, A215734, A215735, A215736, A215737.

Table[k = 1; While[d = IntegerDigits[n^k]; df = Partition[Differences[d], 6, 1]; ! MemberQ[df, {0, 0, 0, 0, 0, 0}],  k++]; d[[Position[df, {0, 0, 0, 0, 0, 0}][[1, 1]]]], {n, 2, 100}] (* T. D. Noe, Oct 02 2012 *)

A217175 a(n) is the first digit (from the left) to appear n times in succession in the decimal representation of the Fibonacci(A217165(n)).

Original entry on oeis.org

0, 5, 7, 7, 1, 5, 7, 7, 3, 2, 4, 3, 4, 2, 4

Offset: 1

V. Raman, Sep 27 2012

Cf. A217165.

Cf. A000045, A215732, A215733, A215734, A215735, A215736, A215737.

k = 0; Join[{0}, Table[While[d = IntegerDigits[Fibonacci[k]]; prt = Partition[Differences[d], n - 1, 1]; ! MemberQ[prt, Table[0, {n - 1}]], k++]; d[[Position[prt, Table[0, {n - 1}]][[1, 1]]]], {n, 2, 8}]] (* T. D. Noe, Oct 03 2012 *)

def A217175(n):
    if n == 1:
        return 0
    else:
        l, y, x = [str(d)*n for d in range(10)], 0, 1
        for m in range(1, 10**9):
            s = str(x)
            for k in range(10):
                if l[k] in s:
                    return k
            y, x = x, y+x
        return 'search limit reached'
# Chai Wah Wu, Dec 17 2014

a(10)-a(11) from Chai Wah Wu, Dec 17 2014

a(12)-a(15) from Nick Hobson, Feb 14 2024

A217176 a(n) is the first digit (from the left) to appear n times in succession in the decimal representation of the Lucas(A217166(n)).

Original entry on oeis.org

2, 1, 3, 0, 2, 2, 9, 7, 2, 1, 1, 5, 5, 7, 7, 9

Offset: 1

V. Raman, Sep 27 2012

Cf. A217166.

Cf. A000032, A215732, A215733, A215734, A215735, A215736, A215737.

k = 0; Join[{2}, Table[While[d = IntegerDigits[LucasL[k]]; prt = Partition[Differences[d], n - 1, 1]; ! MemberQ[prt, Table[0, {n - 1}]], k++]; d[[Position[prt, Table[0, {n - 1}]][[1, 1]]]], {n, 2, 8}]] (* T. D. Noe, Oct 03 2012 *)

def A217176(n):
    if n == 1:
        return 2
    else:
        l, y, x = [str(d)*n for d in range(10)], 2, 1
        for m in range(1, 10**9):
            s = str(x)
            for k in range(10):
                if l[k] in s:
                    return k
            y, x = x, y+x
        return 'search limit reached'
# Chai Wah Wu, Dec 17 2014

a(11) from Chai Wah Wu, Dec 17 2014

a(12)-a(16) from Nick Hobson, Feb 03 2024

cp's OEIS Frontend

A215732 a(n) is the first digit to appear n times in succession in a power of 2.

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A215727 a(n) is the smallest m for which 3^m contains n consecutive identical digits.

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A217171 a(n) is the first digit (from the left) to appear six times in succession in the decimal representation n^A217161(n).

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A217167 a(n) is the first digit (from the left) to appear two times in succession in the decimal representation of n^A217157(n).

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A217168 a(n) is the first digit (from the left) to appear three times in succession in the decimal representation of n^A217158(n).

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A217169 a(n) is the first digit (from the left) to appear four times in succession in the decimal representation of n^A217159(n).

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A217170 a(n) is the first digit (from the left) to appear five times in succession in the decimal representation of n^A217160(n).

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A217172 a(n) is the first digit (from the left) to appear seven times in succession in the decimal representation of n^A217162(n).

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Mathematica

A217175 a(n) is the first digit (from the left) to appear n times in succession in the decimal representation of the Fibonacci(A217165(n)).

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